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Zheng Chen, Jean-Baptiste Caillau, Yacine Chitour
To cite this version:
Zheng Chen, Jean-Baptiste Caillau, Yacine Chitour. L
1-minimization for mechanical systems. SIAM
Journal on Control and Optimization, Society for Industrial and Applied Mathematics, 2016, 54 (3),
pp.1245-1265. �10.1137/15M1013274�. �hal-01136676�
J.-B. Caillau
∗Z. Chen
†Y. Chitour
‡March 2015
Abstract
Second order systems whose drift is defined by the gradient of a given potential are considered, and minimization of the L1-norm of the control is addressed. An analysis of the extremal flow emphasizes the role of sin-gular trajectories of order two [25, 29]; the case of the two-body potential is treated in detail. In L1-minimization, regular extremals are associated with controls whose norm is bang-bang; in order to assess their optimality properties, sufficient conditions are given for broken extremals and related to the no-fold conditions of [20]. An example of numerical verification of these conditions is proposed on a problem coming from space mechanics. Keywords. L1-minimization, second order mechanical systems, order two singular trajectories, no-fold conditions for broken extremals, two-body problem
MSC classification. 49K15, 70Q05
1
Introduction
This paper is concerned with the optimal control of mechanical systems of the following form: ¨ q(t) + ∇qV (q(t)) = u(t) M (t), ˙ M (t) = −β|u(t)|,
where q is valued in an open subset Q of Rm, m ≥ 2, on which the potential V is defined. The second equation describes the variation of the mass, M , of the system when a control is used (β is some nonnegative constant). The finite dimensional norm is Euclidean,
|u| = q
u2
1+ · · · + u2m
∗Math. Institute, Univ. Bourgogne & CNRS/INRIA (jean-baptiste.caillau@u-bourgo
gne.fr). Part of this work was done during a sabbatical leave at Lab. J.-L. Lions, Univ. Paris VI & CNRS, whose hospitality is gratefully acknowledged.
†Math. Dep., Univ. Paris-Sud & CNRS and Northwestern Polytechnical Univ.
(zheng.chen@math.u-psud.fr). Supported by Chinese Scholarship Council (grant no. 2013 0629 0024).
‡L2S-Supelec, Univ. Paris-Sud & CNRS (yacine.chitour@lss.supelec.fr).
and a constraint on the control is assumed,
|u(t)| ≤ ε, ε > 0. (1) Given boundary conditions in the n-dimensional state (phase) space X := T Q ' Q × Rm(n = 2m), the problem of interest is the minimization of consumption, that is the maximization of the final mass M (tf) for a fixed final time. Clearly, this amounts to minimizing the L1-norm of the control,
Z tf
0
|u(t)| dt → min . (2)
Up to some rescaling, there are actually two cases, β = 1 or β = 0. In the second one, the mass is constant; though maximizing the final mass does not make sense anymore, the Lagrange cost (2) is still meaningful. Actually, as propellant is only a limited fraction of the total mass, one can expect this idealized constant mass model to capture the main features of the original problem. We shall henceforth assume β = 0, so the state reduces to x := (q, v) with v := ˙q.
In finite dimensions, `1-minimization is well-known to generate sparse solu-tions having a lot of zero components; this fact translates here into the existence of subintervals of time where the control vanishes, as is clear when applying the maximum principle (see §2). This intuitively goes along well with the idea of minimizing consumption: There are privileged values of the state where the control is more efficient and should be switched on (burn arcs), while there are some others where it should be switched off (cost arcs). (See also [5] for a differ-ent kind of interpretation in a biological setting, again with L1-minimization.) The resulting sparsity of the solution is then tuned by the ratio of the fixed final time over the minimum time associated with the boundary conditions: While a simple consequence of the form of the dynamics (and of the ball constraint on the control) is that the min. time control norm is constant and maximum every-where for the constant mass model,1 the extra amount of time available allows for some optimization that results in the existence of subarcs of the trajectory with zero control. (See Proposition 1, in this respect.) A salient peculiarity of the infinite dimensional setting is the existence of subarcs with intermediate value of the norm of the control, namely singular arcs. This was analyzed in the seminal paper of Robbins [25] in the case of the two-body potential, provid-ing yet another example of the fruitful exchanges between space mechanics and optimal control in the early years of both disciplines. The consequence of these singular arcs being of order two was further realized by Marchal who studied chattering in [18]; this example comes probably second after the historical one of Fuller [12] and has been thoroughly investigated by Zelikin and Borisov in [29, 30].
A typical example of second order controlled system is the restricted three-body problem [8] where, in complex notation (R2' C),
Vµ(t, q) := − 1 − µ |q + µeit|−
µ |q − (1 − µ)eit|·
In this case, µ is the ratio of the masses of the two primary celestial bodies, in circular motion around their common center of mass. The controlled third body
1See, e.g., [10]; this fact remains true for time minimization if the mass is varied provided
is a spacecraft gravitating in the potential generated by the two primaries, but not influencing their motion. When µ = 0, the potential is autonomous and one retrieves the standard controlled two-body problem. The study of ”continuous” (as opposed to impulsive) strategies for the control began in the 60’s; see, e.g., the work of Lawden [16], or Beletsky’s book [3] where the importance of low thrust (small ε in (1)) to spiral out from a given initial orbit was foreseen. There is currently a strong interest for low-thrust missions with, e.g., the Lisa Pathfinder [17] one of ESA2 towards the L1 Lagrange point of the Sun-Earth system, or BepiColombo [4] mission of ESA and JAXA3 to Mercury.
An important issue in optimal control is the ability to verify sufficient op-timality conditions. In L1-minimization, the first candidates for optimality are controls whose norm is bang-bang, switching from zero to the bound prescribed by (1) (more complicated situations including singular controls). Second order conditions in the bang-bang case have received quite an extensive treatment; references include the paper of Sarychev [26], followed by [2] and [19, 22, 23]. On a similar line, the stronger notion of state optimality was introduced in [24] for free final time. More recently, a regularization procedure has been developed in [27] for single-input systems. These papers consider controls valued in poly-hedra; the standing assumptions allow to define a finite dimensional accessory optimization problem in the switching times only. Then, checking a second or-der sufficient condition on this auxiliary problem turns to be sufficient to ensure strong local optimality of the bang-bang controls. A byproduct of the analy-sis is that conjugate times, where local optimality is lost, are switching times. A different approach, based on Hamilton-Jacobi-Bellman and the method of characteristics in optimal control, has been proposed by Noble and Sch¨attler in [20]. Their results encompass the case of broken extremals with conjugate points occuring at or between switching times. We provide a similar analysis by requiring some generalized (with respect to the smooth case) disconjugacy condition on the Jacobi fields, and using instead a Hamiltonian point of view reminiscent of [11, 15]. Treating the case of such broken extremals is crucial for L1-minimization: As the finite dimensional norm of the control involved in the constraint (1) and in the cost (2) is an `2-norm, the control is valued in the Euclidean ball of Rm, not a polyhedron if m > 1. When m = 1, the situation is degenerate, and one can for instance set u = u+− u−, with u+, u− ≥ 0. (This approach also works for m > 1 when an `1 or `∞-norm is used for the values of the control; see, e.g., [28].) When m > 1, it is clear using spherical coordinates that although the norm of the control might be bang-bang, the variations of the control component on Sm−1 preclude the reduction to a finite dimensional optimization problem. (The same remark holds true for any `p-norm of the control values with 1 < p < ∞.) An example of conjugacy occuring between switching times is provided in §4.
The paper is organized as follows. In section 2, the extremal lifts of L1 -minimizing trajectories are studied for an arbitrary potential in the constant mass case; the properties of the flow are encoded by the Poisson structure de-fined by two Hamiltonians. In section 3, sufficient conditions for strong local optimality of broken extremals with regular switching points are given in terms of jumps on the Jacobi fields; these conditions are related to the no-fold
con-2European Space Agency.
dition of [20]. In section 4 some numerical results illustrating the verification of these sufficient conditions for L1-minimizing trajectories are given. The two-body mechanical potential is considered, completing the study of Gergaud and Haberkorn [13] where the first numerical computation of fuel minimizing con-trols with hundreds of switchings (for low thrust) was performed using a clever combination of shooting and homotopy techniques. (See also [21] in the case of a few switchings.) The classical construction of fields of extremals in the smooth case is reviewed in an appendix.
2
Singularity analysis of the extremal flow
By renormalizing the time and the potential, one can assume ε = 1 in (1), so we consider the L1-minimum control of
¨
q(t) + ∇V (q(t)) = u(t), |u(t)| ≤ 1,
with x(t) = (q(t), v(t)) ∈ X = T Q (v(t) = ˙q(t)), Q an open subset of Rm (m ≥ 2), and make the following assumptions on the boundary conditions:
x(0) = x0 and x(tf) ∈ Xf ⊂ X
where (i) x0does not belong to the terminal submanifold Xf, (ii) Xfis invariant wrt. the flow of the drift,4
F0(q, v) = v ∂
∂q − ∇V (q) ∂ ∂v,
and (iii) the fixed final time tf is supposed strictly greater than the minimum time tf(x0, Xf) < ∞ of the problem. As the cost is not differentiable for u = 0, rather than using a non-smooth maximum principle (compare, e.g., [5]) we make a simple desingularization: In spherical coordinates, u = ρw where ρ ∈ [0, 1] and w ∈ Sm−1; the change of coordinates amounts to adding an Sm−1 fiber above the singularity u = 0 of the cost. In these coordinates, the dynamics write
˙ x(t) = F0(x(t)) + ρ(t) m X i=1 wi(t)Fi(x(t))
with canonical Fi = ∂/∂vi, i = 1, . . . , m, and the criterion is linearized: Z tf
0
ρ(t) dt → min . The Hamiltonian of the problem is
H(x, p, ρ, w) = p0ρ + H0(x, p) + ρ m X
i=1
wiψi(x, p)
where H0(x, p) := pF0(x) and the ψi(x, p) := pFi(x) are the Hamiltonian lifts of the Fi, i = 1, . . . , m. Readily, H ≤ H0+ ρH1 with
H1:= p0+ v u u t m X i=1 ψ2 i,
4This assumption can be weakened; it is only used to ensure that a time minimizing control
and the equality can always be achieved for some w ∈ Sm−1: w = ψ/|ψ| whenever ψ := (ψ1, . . . , ψm) is not zero, any w on the sphere otherwise. By virtue of the maximum principle, if (ρ, w) is a measurable minimizing control then the associated trajectory is the projection of an integral curve (x, p) : [0, tf] → T∗X of H0+ ρH1such that, a.e.,
H0(x(t), p(t)) + ρ(t)H1(x(t), p(t)) = max
r∈[0,1]H0(x(t), p(t)) + rH1(x(t), p(t)). (3) Moreover, the constant p0 is nonpositive and (p0, p) 6= (0, 0). Either p0 = 0 (abnormal case), or p0 can be set to −1 by homogeneity (normal case). Proposition 1 (Gergaud et al. [13]). There are no abnormal extremals. Lemma 1. The function ψ evaluated along an extremal has only isolated zeros. Proof. As a function of time when evaluated along an extremal, ψ is absolutely continous and, a.e. on [0, tf],
˙
ψi(t) = p(t)[F0, Fi](x(t)), i = 1, . . . , m.
As a result, ψ is aC1 function of time and, if ψ(t) = 0, then ˙ψ(t) 6= 0; indeed, the rank of {F1, . . . , Fm, [F0, F1], . . . , [F0, Fm]} is maximum everywhere, so p(t) would otherwise be zero. Since p is solution of a linear ode, this would imply that it vanishes identically; necessarily p0< 0, so ρ would also be zero a.e. because of the maximization condition (3). This is impossible because x0∈ X/ f. Proof of the Proposition. By contradiction: Assume p0= 0; as ψ has only iso-lated zeros according to the previous lemma, ρ = 1 a.e. by maximization. The resulting cost is equal to tf. Now, the target submanifold Xf is invariant by the drift, so any minimizing control extended by u = 0 on [tf, tf] (where tf denotes the min. time) remains admissible. It has a cost equal to tf < tf, hence the contradiction.
We set p0 = −1, so H
1 = |ψ| − 1. In contrast with the minimum time case, the singularity ψ = 0 does not play any role in L1-minimization. In the neigh-bourhood of t such that ψ(t) = 0, H1 is negative, so ρ = 0. Locally, the control vanishes and the extremal is smooth. The only effect of the singularity is a discontinuity in the Sm−1fiber over u = 0 in which w(t+) = −w(t−) (see [10]). The important remaining singularity is H1= 0. As opposed to the standard single-input case, H1is not the lift of a vector field on X; the properties of the extremal flow depend on H0, H1, and their Poisson brackets. (See also §4 for the consequences in terms of second order conditions.) We denote by H01 the bracket {H0, H1}, and so forth. The following result is standard (see [6], e.g.) and accounts for the intertwining of arcs along which ρ = 0 (labeled γ0) with arcs such that ρ = 1 (labeled γ+).
Proposition 2. In the neighbourhood of z0 in {H1= 0} such that H01(z0) 6= 0, every extremal is locally bang-bang of the form γ0γ+ or γ+γ0, depending on the sign of H01(z0).
Proof. As H01(z0) 6= 0, H1 must be a submersion at z0, so {H1= 0} is locally a codimension one submanifold splitting T∗X into {H1 < 0} and {H1 > 0}. Evaluated along an extremal, H1is aC1 function of time since
˙
H1(t) = {H0+ ρ(t)H1, H1} = H01(t).
Through z0passes only one extremal, and it is of the form γ0γ+if H01(z0) > 0 (resp. γ+γ0 if H01(z0) < 0). The bracket condition allows to use the implicit function theorem to prove that neighbouring extremals also cross {H1 = 0} transversally.
Such switching points are termed regular and are studied in §3 from the point of view of second order optimality conditions. Besides the occurence of γ0 arcs resulting in the parsimony of solutions as explained in the introduction, the peculiarity of the control setting is the existence of singular arcs along which H1vanishes identically. On such arcs, ρ may take arbitrary values in [0, 1]. Theorem 1 (Robbins [25]). Singular extremals are at least of order two, and minimizing singulars of order two are contained in
{z = (q, v, pq, pv) ∈ T∗X | V00(q)p2v ≥ 0, V 000(q)p3
v> 0}.
Proof. One has H0= (pq|v) − (pv|∇V (q)), and H1= 0 along a singular so, 0 = H01= −
1 |pv|
(pq|pv) along the arc.
Lemma 2. On T∗X, H101= H1001= 0. Proof. Computing, H101= {H1, H01} = {|pv| − 1, − 1 |pv| (pq|pv)} = 0, and it is standard that
H1001 = {H1, {H0, H01}}
= {−H01, H01} + {H0, H101} = 0
using Leibniz rule.
Then 0 = ˙H01= H001+ ρH101 implies H001= 0 along a singular arc. Iterating, 0 = H˙001 = H0001 + ρH1001 so, by the previous lemma again, 0 = H0001. Eventually, 0 = ˙H0001= H00001+ρH10001. Set f := H0, g := H1, h := −(pq|pv), so that H01= βh with β = 1/|pv|. Using Leibniz rule, the following is clear. Lemma 3.
(adkf )(βh)|(adif )h=0, 0≤i<k= β(adkf )h
Computing, one obtains (adf )h = −V00(q)p2v+ |pq|2, so 0 = H001 implies V00(q)p2v≥ 0, and (ad2f )h = −V000(q)(v, pv, pv) + 4V00(q)(pq, pv), {g, (ad2f )h} = − 1 |pv| V000(q)p3v.
Through a point z0 such that the last quantity does not vanish, there passes a so-called order two singular extremal that is an integral curve of the Hamiltonian Hs:= H0+ ρsH1 with the dynamic feedback
ρs:= − H00001 H10001 ·
Along such a minimizing singular arc, the generalized Legrendre condition must hold, H10001≤ 0, so V000(q)p3v> 0.
Corollary 1. In the case of the two-body potential V (q) = −1/|q| (q 6= 0), along an order two singular arc one has either α ∈ (π/2, α0] or α ∈ [−α0, −π/2) where α is the angle of the control with the radial direction, and α0= acos(1/
√ 3). Proof. One has
V0(q)pv= (pv|q) |q|3 , V 00(q)p2 v= |pv|2 |q|3 − 3(pv|q)2 |q|5 , V000(q)p3v= −9(pv|q)|pv| 2 |q|5 + 15(pv|q)3 |q|7 · On Q = Rm\{0}, Sm−1 3 w = p
v since |pv| = 1 along a singular arc, so cos α = (pv|q)/|q|. The condition V00(q)p2v ≥ 0 reads 1 − 3 cos2α ≥ 0, and V000(q)p3
v> 0 is fulfilled if and only if
cos α(3 − 5 cos2α) < 0
that is provided cos α < 0 in addition to the previous condition. Hence the two cases (in exclusion since the singular control is smooth) for the angle.
The existence of order two singular arcs in the two-body case results in the well-known Fuller or chattering phenomenon [18, 29]. The same phenomenon actually persists for the restricted three body problem as is explained in [30]. Altough these singular trajectories are contained in some submanifold of the cotangent space with codimension > 1, their existence rules out the possibil-ity to bound globally the number of switchings of regular extremals described by Proposition 2. The next section is devoted to giving sufficient optimality conditions for such bang-bang extremals.
3
Sufficient conditions for extremals with
regu-lar switchings
Let X be an open subset of Rn, U a nonempty subset of Rm, f a vector field on X parameterized by u ∈ U , and f0: X × U → R a cost function, all smooth. Consider the following minimization problem with fixed final time tf: Find (x, u) : [0, tf] → X × U , x absolutely continous, u measurable and bounded, such that
˙
x(t) = f (x(t), u(t)), t ∈ [0, tf] (a.e), x(0) = x0, x(tf) = xf, and such that
Z tf
0
f0(x(t), u(t)) dt
is minimized. The maximum principle asserts that, if (x, u) is such a pair, there exists an absolutely continuous lift (x, p) : [0, tf] → T∗X and a nonpositive scalar p0, (p0, p) 6= (0, 0), such that a.e. on [0, tf]
˙x(t) = ∂H ∂p(x(t), p(t), u(t)) , ˙p(t) = − ∂H ∂x(x(t), p(t), u(t)) , and H(x(t), p(t), u(t)) = max U H(x(t), p(t), ·) where H : T∗X × U → R is the Hamiltonian of the problem,
H(x, p, u) := p0f0(x, u) + pf (x, u). We first assume that
(A0) The reference extremal is normal.
Accordingly, p0 can be set to −1. Let H1, H2 : T∗X → R be two smooth functions, and denote Σ := {H1= H2}, Ω1:= {H1 > H2} (Ω2:= {H2 > H1}, resp.) We assume that
max
U H(z, ·) = Hi, z ∈ Ωi, i = 1, 2, (4) and follow the point of view of [11] that these two Hamiltonians are competing Hamiltonians. Let (x, p, u) be a reference extremal having only one contact with Σ at z1:= z(t1), t1∈ (0, tf) (z := (x, p)). We denote H12= {H1, H2} the Poisson bracket of H1 with H2 and make the following assumption:
(A1) H12(z1) > 0.
In [15] terms, z1 is a regular (or normal ) switching point. This condition is called the strict bang-bang Legendre condition in [2]. The analysis of this section readily extends to a finite number of such switchings.
Proof. The extremal having only one contact with Σ at z1, z(t) is either in Ω1 or Ω2 for t 6= t1. Because of (4), the maximization condition of the maximum principle implies that z is given by the flow of either H1 or H2 on [0, t1]. In both cases,
d
dt(H1− H2)(z(t))|t=t1= −H12(z1) < 0,
so H1 > H2 before t1 (H2> H1 after t1, resp.) and the only possibility is an H1then H2concatenation of flows.
As a result of (A1), Σ is a codimension one submanifold in the nbd of z1, and one can define locally a function z07→ t1(z0) such that z1(t1(z0), z0) belongs to Σ for z0in a nbd of z0:= z(0). As we have just done, we will denote
zi(t, z0) = et − → Hi(z
0), i = 1, 2,
the Hamiltonian flows of H1 and H2. These flows will be assumed complete for the sake of simplicity. We will denote0 = ∂/∂z for flows. Clearly,
Lemma 5.
t01(z0) =
(H1− H2)0 H12
(z1(t1(z0), z0))z10(t1(z0), z0).
One then defines locally z0 7→ z(t, z0) = (x(t, z0), p(t, z0)) := z1(t, z0) if t ≤ t1(z0), and z(t, z0) := z2(t − t1(z0), z1(t1(z0), z0)) if t ≥ t1(z0). We recall the following standard computation:
Lemma 6. For t > t1(z0), ∂z ∂z0 (t, z0) = z20(t − t1(z0), z1(t1(z0), z0))(I + σ(z0))z10(t1(z0), z0) with σ(z0) = −−−−−−→ H1− H2 (H1− H2)0 H12 (z1(t1(z0), z0)). (5) Proof. The derivative is equal to (arguments omitted)
− ˙z2t01+ z 0
2( ˙z1t01+ z 0 1),
hence the result by factoring out z02 and using Lemma 5 plus the fact that the adjoint action of a flow is idempotent on its generator,
(z20(s, z))−1−→H2(z2(s, z)) = − → H2(z), (s, z) ∈ R × T∗X. The function δ(t) := det ∂x ∂p0 (t, z0), t 6= t1, (6) is piecewise continuous along the reference extremal, and we make the additional assumption that
This condition means that we assume disconjugacy on (0, t1] and [t1, tf] along the linearized flows of H1and H2, respectively, and that the jump (encoded by the matrix σ(z0)) in the Jacobi fields is such that there is no sign change in the determinant. This is exactly the condition one is able to check numerically by computing Jacobi fields (see [6, 7], e.g.). As will be clear from the proof of the result below, geometrically this assumption is the no-fold condition of [20] (no fold outside t1, no broken fold at t1).
Theorem 2. Under assumptions (A0)-(A2), the reference trajectory is a C0 -local minimizer among all trajectories with same endpoints.
Proof. We proceed in five steps.
Step 1. According to (A2), ∂x1/∂p0(t, z0) is invertible for t ∈ (0, t1]; one can then construct a Lagrangian perturbation L0 transverse to Tx∗0X containing
z0 such that ∂x1/∂z0(t, z0) is invertible for t ∈ [0, t1], t = 0 included, ∂/∂z0 denoting the n partials wrt. z0∈L0. (See appendix A.) For ε > 0 small enough define
L1:= {(t, z) ∈ R × T∗X | (∃z0∈L0) : t ∈ (−ε, t1(z0) + ε) s.t. z = z1(t, z0)}. By restrictingL0if necessary, Π : R × T∗X → R × X, (t, z) 7→ (t, x) induces a diffeomorphism ofL1 onto its image. Similarly, (A2) implies that
∂ ∂p0
[x2(t − t1(z0), z1(t1(z0), z0))] |z0=z0
is invertible for t ∈ [t1, tf]; restricting again L0 if necessary, one can assume that Π also induces a diffeomorphism from
L2:= {(t, z) ∈ R × T∗X | (∃z0∈L0) : t ∈ (t1(z0) − ε, tf+ ε) s.t. z = z2(t − t1(z0), z1(t1(z0), z0))}
onto its image.
Step 2. Define Σ1:=L1∩(R×Σ). As (t, z0) 7→ (t, x1(t, z0)) is a diffeomorphism from R ×L0 onto Π(L1), there exists an inverse function z0(t, x) such that Π(Σ1) = {ψ = 0} with
ψ(t, x) := t − t1(z0(t, x)).
Denote ψ(t) := ψ(t, x(t)) the evaluation of this function along the reference trajectory. By construction, ˙ψ(t1−) = 1 > 0 and (compare [20])
˙ ψ(t1+) = 1 + ∂t1 ∂z0 (z0) ∂x1 ∂z0 (t1, z0) −1 ∇p(H1− H2)(z1). Lemma 7. δ(t1+) = δ(t1−) 1 + ∂t1 ∂p0 (z0) ∂x1 ∂p0 (t1, z0) −1 ∇p(H1− H2)(z1) ! . (7)
Proof. By virtue of Lemma 6, ∂x ∂p0 (t1+, z0) = ∂x1 ∂p0 (t1, z0) + ∇p(H1− H2) (H1− H2)0 H12 (z1) ∂z1 ∂p0 (t1, z0) | {z } = ∂t1 ∂p0 (z0)
(the second equality coming from Lemma 5). Assumption (A2) implies δ(t1−) 6= 0 so, taking determinants,
δ(t1+) = δ(t1−) det I + ∂x1 ∂p0 (t1, z0) −1 ∇p(H1− H2)(z1) ∂t1 ∂p0 (z0) ! = δ(t1−) 1 + ∂t1 ∂p0 (z0) ∂x1 ∂p0 (t1, z0) −1 ∇p(H1− H2)(z1) ! as det(I + xty) = 1 + (x|y).
Since δ(t1+) and δ(t1−) have the same sign, the quantity in brackets in (7) must be positive. Accordingly, ˙ψ(t1+) > 0 asL0 can be taken arbitrarily close to Tx∗0X. So, locally, Π(Σ1) is a submanifold that splits R × X in two and, by restrictingL0 if necessary, every extremal of the field t 7→ x(t, z0) for z0∈L0 crosses Π(Σ1) transversally. Defining
L− 1 := {(t, z) ∈ R × T∗X | (∃z0∈L0) : t ∈ [0, t1(z0)] s.t. z = z1(t, z0)} and L+ 2 := {(t, z) ∈ R × T ∗X | (∃z 0∈L0) : t ∈ [t1(z0), tf] s.t. z = z2(t − t1(z0), z1(t1(z0), z0))},
one can hence piece together the restrictions of Π toL1− andL2+into a contin-uous bijection fromL1−∪L+
2 into Π(L − 1 ∪L
+
2 ). By restricting to a compact neighbourhood of the graph of z, one may assume that Π induces a homeomor-phism on its image.
Step 3. Denote αi := p dx − Hi(z)dt, i = 1, 2, the Poincar´e-Cartan forms associated with H1 and H2, respectively. To prove that α1 is exact on L1, it is enough to prove that it is closed. Indeed, if γ(s) := (t(s), z1(t(s), z0(s))) is a closed curve on L1, it retracts continuously on γ0(s) := (0, z0(s)) so that, provided α1 is closed, Z γ α1= Z γ0 α1= Z γ0 p dx = 0
because z0(s) belongs to L0 that can be chosen such that p dx is exact on it. (Compare [1, §17].) Similarly, to prove that α2 is exact on L2, it suffices to prove that it is closed: If γ(s) := (t(s), z2(t(s) − t1(z0(s)), z1(t1(z0(s)), z0(s)))) is a closed curve in L2, it readily retracts continuously on the curve γ1(s) := (t1(z0(s)), z1(t1(z0(s)), z0(s))) in Σ1, which retracts continuously on γ0(s) := (0, z0(s)) again. Then, as H1= H2 on Σ, Z γ α2= Z γ1 α2= Z γ1 α1= Z γ0 α1
Figure 1: The field of extremals.
that vanishes as before. To prove that α1 is closed, consider tangent vectors at (t, z) ∈L1; a parameterization of this tangent space is
(δt,−→H1(z)δt + z01(t, z0)δz0), (δt, δz0) ∈ R × Tz0L0
where z0∈L0 is such that z = z1(t, z0). For two such vectors v1, v2, dα1(t, z)(v1, v2) = (dp ∧ dx − dH1(z)dt)(v1, v2)
= dp ∧ dx(z01(t, z0)δz10, z01(t, z0)δz20) = dp ∧ dx(δz10, δz02)
because exp(t−→H1) is symplectic and L0 is Lagrangian. Regarding α2, the tan-gent space at (t, z) ∈L2is parameterized according to
(δt,−→H2(z)δt + z02(t − t1(z0), z1(t1(z0), z0))(I + σ(z0))z10(t, z0)δz0) with (δt, δz0) ∈ R × Tz0L0, and where z0 ∈ L0 is such that z = z2(t −
t1(z0), z1(t1(z0), z0)). For two such vectors v1, v2, dα2(t, z)(v1, v2) = (dp ∧ dx − dH2(z)dt)(v1, v2)
= dp ∧ dx((I + σ(z0))z01(t, z0)δz10, (I + σ(z0))z01(t, z0)δz20) = dp ∧ dx(z01(t, z0)δz10, z01(t, z0)δz20)
because exp(t−→H2) is symplectic and because Lemma 8.
I + σ(z0) ∈ Sp(2n, R). Proof. For any z ∈ R2n,
t(I + J ztz)J (I + J ztz) = J − ztz + ztz + z(tzJ z | {z } 0
)tz = J.
This proves the lemma because of the definition (5) of σ(z0).
One then concludes as before that α2 is closed using the fact that exp(t − → H1) is symplectic andL0 is Lagrangian.
Step 4. Let (x, u) : [0, tf] → X × U be an admissible pair. We first assume that x is of class C1 and that its graph has only one isolated contact with Π(Σ1), at some point point (t1, x(t1)). For x close enough to x in the C0-topology, this graph has a unique lift t 7→ (t, x(t), p(t)) in L1−∪L2+. As a gluing at t1 of two absolutely continous functions, z := (x, p) : [0, tf] → T∗X is absolutely continous. Denote γ1 and γ2the two pieces of this lift. Denote similarly γ1 and γ2the pieces of the graph of the extremal z (see Fig. 2). One has
Z tf 0 f0(x(t), u(t)) dt = Z t1 0 + Z tf t1 (p(t) ˙x(t) − H(x(t), p(t), u(t))) dt ≥ Z t1 0 (p(t) ˙x(t) − H1(x(t), p(t))) dt + Z tf t1 (p(t) ˙x(t) − H2(x(t), p(t))) dt = Z γ1 α1+ Z γ2 α2
since z(t) belongs to Ω1for t ∈ [0, t1) (resp. to Ω2for t ∈ (t1, tf]). By connected-ness, there exists a smooth curve γ12⊂ Σ1 connecting (t1, z(t1)) to (t1, z(t1));
Figure 2: Integration paths.
having the same endpoints, γ1 and γ1∪ γ12(resp. γ2 and −γ12∪ γ2) are homo-topic. Since α1 and α2 are exact one forms onL1and L2, respectively,
Z γ1 α1+ Z γ2 α2 = Z γ1∪γ12 α1+ Z −γ12∪γ2 α2 = Z γ1 α1+ Z γ2 α2 = Z tf 0 f0(x(t), u(t)) dt since H1= H2 on Σ.
Step 5. Consider finally an admissible pair (x, u), x close enough to x in the C0-topology. One can find
e
x of class C1 arbitrarily close to x in the W1,∞ -topology such that ex(0) = x0 and ex(tf) = xf. Moreover, as Π(Σ1) is a locally a smooth manifold, up to someC1-small perturbation one can assume that the graph of ex has only transverse intersections with Π(Σ1). Let ez := (x,e p) denotee the associated lift; one has
f0(x(t), u(t)) = (e p(t) ˙e x(t) − H(e ex(t),p(t), u(t))) +e p(t)(f (e ex(t), u(t)) − ˙x(t)),e and the second term in the right-hand side can be made arbitrarily small when e
x gets closer to x in the W1,∞-topology since (t, e
z(t)) = Π−1(t, e
x(t)) remains bounded by continuity of the inverse of Π. Let then ε > 0; as a result of the previous discussion, there exists ex of class C1 with same endpoints as x and whose graph has only isolated contacts with Π(Σ1) such that
Z tf 0 f0(x(t), u(t)) dt ≥ Z tf 0 f0(x(t), u(t)) dt − ε,e
Z tf 0 f0(x(t), u(t)) dt ≥e Z tf 0 (p(t) ˙e x(t) − H(e x(t),e p(t), u(t))) dt − ε.e One can extend straightforwardly the analysis of the previous step to finitely many contacts with Π(Σ1), and bound below the integral in the right-hand side of the second inequality by the cost of the reference trajectory. As ε is arbitrary, this allows to conclude.
4
Numerical example: The two-body potential
Following [13], we consider the two-body controlled problem in dimension three. Restricting to negative energy, orbits of the uncontrolled motion are ellipses, and the issue is to realize minimum fuel transfer between non-coplanar orbits around a fixed center of mass. The potential is V (q) := −µ/|q| defined on Q := {q ∈ R3 | q 6= 0}, and we actually restrict to
X := {(q, v) ∈ T Q | |v|2/2 − µ/|q| < 0, q ∧ v > 0}.
(The last condition on the momentum avoids collisional trajectories and ori-entates the elliptic orbits.) The constant µ is the gravitational constant that depends on the attracting celestial body. To keep things clear, a medium thrust case is presented below; the final time is fixed to 1.3 times the minimum time, approximately, which already ensures a satisfying gain of consumption [13]. In order to have fixed endpoints to perform a conjugate point test according to §3 result, initial and final positions are fixed on the orbits (fixed longitudes5). A more relevant treatment would leave the final longitude free (in accordance with assumption (ii) on the target in §2); this would require a focal point test that could be done much in the same way (see, e.g., [9]). See Tab. 1 for a summary of the physical constants.
As explained in §2, the L1-minimization results in a competition between two Hamiltonians: H0 (coming from the drift, only), and H0+ H1 (assuming the control bound is normalized to 1 after some rescaling). Both Hamiltonians are smooth and fit in the framework set up in §3 to check sufficient optimality conditions. Restricting to bang-bang (in the norm of the control) extremals, regularity of the switchings is easily verified numerically, while normality is taken care of by Proposition 1. Then one has to check the no-fold condition on the Jacobi fields. The optimal solution (see Fig. 3) and these fields are computed using the hampath software [14]; as in [9, 13], a regularization by homotopy is used to capture the switching structure and initialize the computation of the bang-bang extremal by single shooting. We are then able to check condition (A2) directly on this extremal by a simple sign test (including the jumps on the Jacobi fields at the regular switchings) on the determinant of the fields (see Fig. 4). An alternative approach would be to establish a convergence result as in [27], and to verify the second order conditions on the sequence of regularized extremals. As underlined in §1 and §3, conjugate times may occur at or between
5Precisely, the longitude l is defined as the sum of three broken angles: l = Ω + θ + $,
where Ω is the longitude of the ascending node (first Euler angle of the orbit plane with the equatorial plane; the second Euler angle defines the inclination of the orbit), θ is the argument of perigee (angle of the semi-major axis of the ellipse, equal to the third Euler angle of the orbit plane), and $ is the true anomaly (polar angle with respect to the semi-major axis in the orbit plane). Here, Ω = θ = 0 on the initial and final orbits.
Table 1: Summary of physical constants used for the numerical commputation. Gravitational constant µ of the Earth: 398600.47 Km3s−2 Mass of the spacecraft: 1500 Kg Thrust: 10 Newtons Initial perigee: 6643 Km Final perigee: 42165 Km Initial apogee: 46500 Km Final apogee: 42165 Km Initial inclination: 0.1222 rad Final inclination: 0 rad Initial longitude: π rad Final longitude: 56.659 rad Minimum time: 110.41 hours Fixed final time: 147.28 hours L1cost achieved (normalized): 67.617
switching times. On the example treated, no conjugate point is detected on [0, tf], ensuring strong local optimality. The extremal is then extended up to 3.5 tf, and a conjugate point is detected about 3.2 tf, at a switching point (sign change occurint at the jump). A second test is provided Fig. 5; by perturbing slightly the endpoint conditions, one observes that conjugacy occurs not at a switching anymore, but along a burn arc.
Remark 1. As H0 is the lift of a vector field, the determinant of Jacobi fields is either identically zero or non-vanishing along a cost arc (ρ = 0). (Compare with the case of polyhedral control set; see also Corollary 3.9 in [20].) Moreover, coming from a mechanical system, the drift F0is the symplectic gradient of the energy function,
E(q, v) := 1 2|v|
2+ V (q).
Accordingly, the δx = (δq, δv) part of the Jacobi field (see appendix) along an integral arc of −→H0verifies
δ ˙x(t) =−→E0(x(t))δx(t),
so δx has a constant determinant along such an arc since the associated flow is symplectic. In particular, the disconjugacy condition (A2) implies that the optimal solution starts with a burn arc.
Conclusion
We have reviewed some of the particularities of L1-minimization in the control setting. Among these, the existence of singular controls valued in the interior of the Euclidean ball comes in strong contrast with the finite dimensional case. Moreover, these singular extremals are at least of order two, entailing existence of chattering [29]. By changing coordinates on the control, one can reduce the system to a single control, namely the norm of the original one. This emphasizes the role played by the Poisson structure of two Hamiltonians, the second not the lift of a vector field; this fact accounts for the possibility of conjugacy happening
Figure 3: L1minimum trajectory. The graph displays the trajectory (blue line), as well as the action of the control (red arrows). The initial orbit is strongly eccentric (0.75) and slightly inclined (7 degrees). The geostationary target orbit around the Earth is reached at tf ' 147.28 hours. The sparse structure of the control is clearly observed, with burn arcs concentrated around perigees and apogees (see [13]). The minimization leads to thrust only 46% of the time.
not necessarily at switching times, as opposed to the simpler case of bang-bang controls valued in polyhedra. Sufficient conditions for this type of extremals have been given; they rely on a simple and numerically verifiable check on the discontinuous Jacobi fields of the system. They are essentially equivalent to the no-fold conditions of [20], formulated here in a Hamiltonian setting. The exam-ple of L1-minimization for the three-dimensional two-body potential illustrates the interest of the approach. Future work include the treatment of mass varying systems (that is of maximization of the final mass) for more general problems such as the restricted three-body one.
A
Sufficient conditions in the smooth case
Consider the same minimization problem as in §3. Suppose that (B0) The reference extremal is normal.
Having fixed p0 to −1, we make the stronger assumption that the maximized Hamiltonian is well defined and smooth, and set
h(z) := max
U H(z, ·), z ∈ T ∗X.
Scholium. For almost all t ∈ [0, tf], h0(z(t)) = ∂H
∂z (z(t), u(t)), ∇
2h(z(t)) − ∇2
zzH(z(t), u(t)) ≥ 0. Proof. For a.a. t ∈ [0, tf], h(z(t)) − H(z(t), u(t)) = 0, while
h(z) − H(z, u(t)) ≥ 0, z ∈ T∗X,
by definition of h. Applying the first and second order necessary conditions for optimality on T∗X at z = z(t) gives the result.
0 100 200 300 400 500 #10-8 -4 -2 0 2 475 476 477 478 #10-13 -1 -0.5 0 0.5 1 1.5 0 50 100 150 200 250 300 350 400 450 500 0 0.2 0.4 0.6 0.8 1
Figure 4: Conjugate point test on the bang-bang L1-extremal extended to [0, 3.5 tf]. The value of the determinant of Jacobi fields (6) along the extremal is plotted against time on the upper left subgraph. The first conjugate point occurs at t1c' 475.93 hours > tf; optimality of the reference extremal on [0, tf] follows. On the upper right subgraph, a zoom is provided to show the jumps on the Jacobi fields (then on their determinant) around the first conjugate time; several jumps are observed, the first one leading to a sign change at the con-jugate time. Note that in accordance with Remark 1, the determinant must be constant along the cost arcs (ρ = 0) provided the symplectic coordinates x = (q, v) are used; this is not the case here as the so called equinoctial ele-ments [10] are used for the state—hence the slight change in the determinant. The bang-bang norm of the control, rescaled to belong to [0, 1] and extended to 3.5 tf, is portrayed on the lower graph. On the extended time span, there are already more than 70 switchings though the thrust is just a medium one. For low thrusts, hundreds of switchings occur.
We make the following assumption on the smooth reference extremal. (B1) ∂x/∂p0(t, z0) is invertible for t ∈ (0, tf].
Theorem 3. Under assumptions (B0)-(B1), the reference trajectory is a C0 -local minimizer among all trajectories with same endpoints.
0 200 400 #10-8 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 489 490 491 #10-14 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5
Figure 5: Conjugate point test on a perturbed bang-bang L1-extremal extended to [0, 3.5 tf]. The value of the determinant of Jacobi fields (6) along the extremal is plotted against time (detail on the right subgraph). The endpoint conditions x0, xf given in Tab. 1 are perturbed according to x ← x + ∆x, |∆x| ' 1e − 5, leading to conjugacy not at but between switching points—along a burn arc (ρ = 1). The first conjugate point occurs at t1c ' 489.23 hours > tf, ensuring again optimality of the reference extremal on [0, tf].
Note that no Legendre type assumption is made, and that the disconjugacy con-dition (B1) can be numerically verified (e.g., by a rank test while integrating the variational system along the reference extremal). For the sake of completeness, we provide a proof that essentially goes along the lines of [1, §21].
Proof. For S0 symmetric of order n, L0 := {δx0 = S0δp0} is a Lagrangian subspace of Tz0(T
∗X). Denote by δz = (δx, δp) the solution of the linearized system
δ ˙z(t) =−→h0(z(t))δz(t), δz(0) = (S0, I),
and set δz(t) = (δe ex(t), δp(t)) := Φe −1t δz(t) where Φtis the fundamental solution of the linearized system
˙ Φt= ∂−→H ∂z (z(t), u(t))Φt, Φ0= I. As δp(0) = δp(0) = I,e S(t) := δx(t)δe p(t)e −1 is well defined for small enough t ≥ 0. Since
Lt:= exp(t − →
h )0(z(t))(L0) and Φ−1t (Lt)
are Lagrangian as images of L0through linear symplectic mappings, S(t) must be symmetric.
Proof. Let t1≥ 0 such that S(t1) is well defined, and let ξ ∈ Rn. Set ξ0:= δp(te 1)−1ξ and δze1(t) := δz(t)ξe 0.
Then δez1(t1) = (S(t1)ξ, ξ), and δxe1(t) = S(t)δpe1(t). Differentiating the previous relation and using S(t) symmetry leads to
( ˙S(t)δpe1(t)|δpe1(t)) = ω(δze1(t), δ ˙ez1(t)). Differentiating now δze1(t) = Φ−1t δz(t)ξ0, one gets δ ˙ze1(t) = Φ−1t ( − → h0(z(t)) −∂ − → H ∂z (z(t), u(t)))Φtδze1(t) = JtΦt(∇2h(z(t)) − ∇2zzH(z(t), u(t)) | {z } ≥ 0 )Φtδze1(t).
(J denotes the standard symplectic matrix.) Evaluating at t = t1, one eventually gets ( ˙S(t1)ξ|ξ) ≥ 0.
For S0= 0, there is η > 0 such that S(t) is well defined on [0, η], which remains true for S0 > 0, |S0| small enough. By the lemma before, St> 0 on [0, η]. In particular, it is an invertible matrix, which ensures that Φ−1t (Lt) is transversal to ker π0(z
0) (π : T∗X → X being the canonical projection), that is Lt is transversal to ker π0(z(t)) by virtue of
Scholium. Φt(ker π0(z0)) = ker π0(z(t))
Proof. Note that in the linearized system defining Φt, δ ˙x(t) = ∇2
xpH(z(t), u(t))δx(t), δ ˙p(t) = −∇2
xxH(z(t), u(t))δx(t) − ∇2pxH(z(t), u(t))δp(t), the equation on δx is linear. Hence δx(0) = 0 implies δx ≡ 0.
By restricting if necessary |S0|, (B1) allows to assume that δx(t) remains in-vertible for t ∈ [η, tf], so transversality of Lt holds on [0, tf]. As a result, one can devise a Lagrangian submanifold L0 of T∗X whose tangent space at z0 is L0; then L := {(t, z) ∈ R × T∗X | (∃z 0∈L0) : t ∈ (−ε, tf+ ε) s.t. z = exp(t − → h )(z0)} is well defined for ε small enough, and such that Π : R × T∗X → R × X induces a diffeomorphism from L onto its image. One can moreover choose L0 such that p dx is not only closed but an exact form on it, in order that the Poincar´ e-Cartan form p dx − h(z)dt is exact onL . This, together with assumption (B0), allows to conclude as usual that the reference trajectory is optimal with respect toC0-neighbouring trajectories with same endpoints.
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![Figure 4: Conjugate point test on the bang-bang L 1 -extremal extended to [0, 3.5 t f ]](https://thumb-eu.123doks.com/thumbv2/123doknet/13396509.405920/19.892.194.700.191.689/figure-conjugate-point-test-bang-bang-extremal-extended.webp)
![Figure 5: Conjugate point test on a perturbed bang-bang L 1 -extremal extended to [0, 3.5 t f ]](https://thumb-eu.123doks.com/thumbv2/123doknet/13396509.405920/20.892.193.698.193.453/figure-conjugate-point-test-perturbed-bang-extremal-extended.webp)
